I broke both of those in the BMC I owned. Can't remember how I got it out without breaking the mirror but I do remember that it was tedious and nerve wracking. I wasn't going to go through that again so I did some Texas engineering.
It seems that I remember there were two threaded holes in the metal housing. I drilled thru the wooden back of the door and put stainless screws with fender washers in from the back side.
I can't remember all the details but it seems like that I might have needed to thread the metal holes to accept the screws from the back.

I like Mark B's idea of removing one of the door catches. Those doors are way too hard to open. The only bad part of that would be the twisting effect on the door as it is pulled open.

Mike you are correct they thread in to loosen and out to tighten. My LXI has them on the top and bottom of the door. I have gotten the old handle out yet however I'm still perplexed as the old handle had a spring in it to keep the handle closed. The replacement ones dont. I have the spring bot have no Idea how they put it in the handle. The things on these birds just keep ya saying hmmmmm